Do you mean a particle is restricted to move on a torus surface (inside a torus and always in contact with the surface)?
What is the interaction force between particle and the torus surface? Normal force ?
or Is there another force to keep particle always in contact with the torus surface (it might fall down due to gravity).
Assume the larger radius is R, and smaller radius is r for the torus.
Does , and in your case?
I do not understand the meaning of in the attached picture.
Your problem might be similar to a circular motion in vertical direction , plus a constant angular speed rotational motion (angular momentum is conserved).